Teichmüller curves in genus two:
The decagon and beyond
Curtis T. McMullen∗
30 April, 2004
Contents
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2
3
4
5
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7
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9
1
Introduction . . . . . . . . . . .
Elementary moves . . . . . . .
Prototypical splittings . . . . .
Sifting . . . . . . . . . . . . . .
Aperiodicity . . . . . . . . . . .
Finiteness . . . . . . . . . . . .
Characterization of Teichmüller
Curves with D = 5 . . . . . . .
Curves with D ≤ 400 . . . . . .
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1
6
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15
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23
Introduction
Let Mg denote the moduli space of Riemann surfaces of genus g, and ΩMg →
Mg the bundle of Abelian differentials. A point in ΩMg is specified by a pair
(X, ω), where X is in Mg and where ω ∈ Ω(X) a nonzero holomorphic 1-form
on X.
There is a natural action of SL2 (R) on ΩMg , giving moduli space a dynamical flavor. The projection of any orbit SL2 (R) · (X, ω) yields a holomorphic
Teichmüller disk f : H → Mg , whose image is typically dense. On rare occasions, however, the stabilizer SL(X, ω) of the given form is a lattice in SL2 (R);
then the image of the quotient map
f : V = H/ SL(X, ω) → Mg
is an algebraic curve, isometrically embedded for the Teichmüller metric.
In this paper we address the classification of such Teichmüller curves in the
case g = 2.
Billiards. Let P ⊂ C be a polygon with angles in πQ. Via an unfolding
construction, (P, dz) determines a holomorphic 1-form (X, ω) such that billiard
trajectories in P correspond to geodesics on (X, |ω|).
∗ Research
partially supported by the NSF.
2000 Mathematics Subject Classification: Primary 32G15, Secondary 37D50, 11F41.
1
We say P is a lattice polygon if SL(X, ω) is a lattice; equivalently, if (X, ω)
generates a Teichmüller curve. In this case, Veech showed that SL(X, ω) allows
one to renormalize the geodesic flow on (X, |ω|), and thereby establish optimal
dynamical properties for billiards in P : for example,
1. Every billiard trajectory is either periodic or uniformly distributed; and
2. The number of types of closed trajectories of length ≤ L is asymptotic to
CP · L2 as L → ∞.
Veech also showed that every regular polygon is a lattice polygon [V1]. In
particular the regular pentagon, octagon and decagon give rise to Teichmüller
curves in genus two.
Hilbert modular surfaces. The geometry of Teichmüller curves as above is
best understood in the case of genus two: any such curve lies on a unique Hilbert
modular surface HD , D > 0 [Mc1]. More precisely, we have a commutative
diagram
f
V −−−−→ M2




y
yJac
HD −−−−→ A2 ,
where HD = (H × H)/ SL2 (OD ) parameterizes the locus of Abelian surfaces
A ∈ A2 with real multiplication by the quadratic order OD ∼
= Z[x]/(x2 + bx+ c),
D = b2 − 4c. We refer to D as the discriminant of the Teichmüller curve
f : V → M2 .
Elliptic differentials. If a billiard table can be tiled by squares, then it is a
lattice polygon. Similarly, for any covering
p : X → E = C/Λ
branched over torsion points on an elliptic curve E, if we set ω = p∗ (dz) then
SL(X, ω) is a lattice, commensurable to SL2 (Z). Thus (X, ω) generates a Teichmüller curve, providing examples in every moduli space Mg .
For d > 1, the surface Hd2 carries infinitely many Teichmüller curves, all
inherited from genus one
√ as above. Our main result shows the situation is
radically different when D is irrational.
Theorem 1.1 (Finiteness) If D is not a square, then there are only finitely
many Teichmüller curves of discriminant D.
We remark that HD always carries infinitely many Shimura curves, covered
g
by graphs of Möbius transformations A : H → H in H
D . In contrast, the
Teichmüller curves in HD are covered by graphs of transcendental
functions
√
[Mc1, §10], and their abundance depends on the rationality of D.
Primitive curves. A Teichmüller curve is primitive if it does not arise from
a curve of lower genus via a branched covering construction. In genus two, a
2
√
Teichmüller curve of discriminant D is primitive if and only if D is irrational.
Although the surface Hd2 carries infinitely many Teichmüller curves, none of
them are primitive.
To construct primitive examples, let the Weierstrass curve WD be the locus
of those Riemann surfaces X ∈ M2 such that
(i) Jac(X) admits real multiplication by O D , and
(ii) X carries an eigenform ω with a double zero at one of the six Weierstrass
points of X.
(Here ω ∈ Ω(X) is an eigenform if OD ·ω ⊂ C · ω.)
It can be shown that WD has either one or two irreducible components, each
of which is a Teichmüller curve of discriminant D generated by billiards in an
L-shaped table. Conversely, every Teichmüller curve generated by a form with
a double zero belongs to some WD [Mc1], [Mc3].
The regular decagon. The curves W5 and W8 are irreducible; they are exactly
the Teichmüller curves generated by billiards in the regular pentagon and the
regular octagon.
On the other hand, the vertices of the regular decagon fall into two equivalence classes when opposite edges are identified. It follows that the corresponding Teichmüller curve is generated by a form with a pair of simple zeros, rather
than a single double zero. We suspect this is the only such example.
Conjecture 1.2 The regular decagon gives the only primitive Teichmüller curve
V → M2 generated by a form with simple zeros.
This conjecture implies:
1. All Teichmüller curves generated by forms of genus two are already known:
they come from branched covers of tori [GJ], billiards in L-shaped tables
[Mc3], and billiards in the regular decagon [V1].
2. For nonsquare discriminant D > 5, the only Teichmüller curves with discriminant D are the components of the Weierstrass curve WD .
(We remark that the curves generated by the regular pentagon and octagon are
also generated by suitable L-shaped tables.)
Algorithms. The proof of Theorem 1.1 is constructive, and it yields an effective algorithm to list all the Teichmüller curves of a given discriminant D. As
evidence for Conjecture 1.2, in §9 we describe the proof of:
Theorem 1.3 The conjecture above holds for all Teichmüller curves with discriminant D ≤ 400.
In particular, in §8 we show:
Theorem 1.4 There are only two Teichmüller curves of discriminant D = 5:
one generated by the regular pentagon, and one by the regular decagon.
3
Similarly, billiards in the regular octagon gives the unique Teichmüller curve
with D = 8; the unique curve with D = 12 is generated by the polygon shown
in Figure 1; and every other primitive Teichmüller curve with D ≤ 400 comes
from billiards in an L-shaped table, given explicitly in [Mc3, Cor. 1.3].
1
λ
1
λ
Figure 1. Billiard table for the unique Teichmüller
curve of discriminant D = 12;
√
λ = (1 + 3)/2.
Proof of finiteness. We turn to a sketch of the proof of Theorem 1.1. Our
goal is to √
show there are only finitely many Teichmüller curves of discriminant
D, when D is irrational.
1. First some definitions. A splitting prototype of discriminant D is a set of
integers p = (a, b, c, e) such that
D = e2 + 4bc,
0 < b,
0 ≤ a < gcd(b, c),
0 < c, and
c + e < b,
gcd(a, b, c, e) = 1.
There are only finitely many prototypes for a given discriminant D.
√
2. Let λ = (e + D)/2 and let It = [0, tλ], where t ∈ (0, 1]. Let (Ei , ωi ) =
(C/Λi , dz), i = 1, 2 be the forms of genus one with period lattices Λ1 =
Z(λ, 0) ⊕ Z(0, λ) and Λ2 = Z(b, 0) ⊕ Z(a, c).
The prototypical form of type p and width t is given by the connected sum
(Xt , ωt ) = (E1 , ω1 )#(E2 , ω2 ).
It
The connected sum is obtained by slitting each torus open along the projection of the arc It , and then gluing corresponding edges (§3). The form
(Xt , ωt ) has a double zero when t = 1, and otherwise a pair of simple
zeros.
3. Let f : V → M2 be a Teichmüller curve of discriminant D. In §3 we show
that any such curve is generated by a prototypical form. Thus the search
for Teichmüller curves is reduced, for each prototype p, to the study of
suitable values of the width parameter t.
4. Any form of genus two can be presented as a connected sum in infinitely
many ways. In §4 we show that any homology class
C ∈ H1 (E1 , Z) ⊕ H1 (E2 , Z) ∼
= H1 (Xt , Z)
4
determines an open interval U (C) ⊂ (0, 1) such that for all t ∈ U (C), we
have a second splitting
(Xt , ωt ) = (F1 , η1 )#(F2 , η2 )
J
with [J] − [I] = C. We also construct a countable set T (C) ⊂ R ∪ {∞},
projectively equivalent to P1 (Q), such that if t ∈ U (C) and SL(Xt , ωt ) is
a lattice, then we also have t ∈ T (C). (The set T (C) is determined by the
condition that the slope of J agrees with the slope of a period of (F1 , η1 ).)
5. In §5 we show that for each prototype p, there are infinitely many homology
√ classes C such that U (C) contains a neighborhood of t = 0. When
D is irrational, we can also insure that T (C1 ) ∩ T (C2 ) is finite for two
such classes. Thus we can find a t0 (D) > 0 such that:
(∗) t < t0 =⇒ SL(Xt , ωt ) is not a lattice.
6. RLet Ω1 ED denote the set of eigenforms of discriminant D normalized by
2
X |ω| = 1. In [Mc5] we analyze the dynamics of SL2 (R) on forms of
genus two, and show in particular that any orbit
Z = SL2 (R) · (X, ω) ⊂ Ω1 ED
is either closed or dense. We also show the closed orbits in Ω1 ED are
isolated: only finitely many meet any compact set.
7. Now suppose there are infinitely many Teichmüller curves Vi of discriminant D. Then there are infinitely many closed orbits Zi = Ω1 Vi ⊂ Ω1 ED .
Each is generated by a prototypical form of type pi and width ti . Passing to a subsequence, we can assume pi = p is constant and ti ∈ (0, 1] is
convergent. But the orbits Zi are isolated, so the only possible limit of ti
is t = 0. This contradicts (∗), and therefore the number of Teichmüller
curves of discriminant D is finite.
Converse Veech dichotomy. Are there billiards with optimal dynamics that
cannot be analyzed via Teichmüller curves? In §7 we use the same methods to
show the answer is no in genus two. Namely, we have:
Theorem 1.5 Let (X, ω) be a holomorphic 1-form of genus two. Then the
following conditions are equivalent:
1. The group SL(X, ω) is a lattice in SL2 (R).
2. For every s ∈ R ∪ {∞}, the foliation of (X, |ω|) by geodesics of slope s is
either periodic or uniquely ergodic.
The implication (1) =⇒ (2) is the well-known Veech dichotomy, valid in any
genus [V1]; the result above furnishes a converse in genus two.
5
Higher genus. In contrast to the case of genus two, at present only finitely
many primitive Teichmüller curves are known for each genus g ≥ 3. Of these, the
Veech examples coming from regular polygons also arise in Mestre’s construction
of families of curves with real multiplication [Me], and Möller shows the Jacobian
of X always has special endomorphisms when (X, ω) generates a Teichmüller
curve [Mo]. Thus the theory of real multiplication may facilitate a classification
in higher genus, as it does in genus two.
Notes and references. In [Mc5] we classify the orbit closures and ergodic invariant measures for the action of SL2 (R) on the space ΩM2 . The classification
is explicit, apart from the issue of describing all Teichmüller curves in genus
two. The present paper and [Mc3] undertake this description.
For additional background on Teichmüller curves, see [Th], [V1], [V2], [Vo],
[Wa], [KS], [Pu], [GJ], [EO], [Lei] and [Lo]. Further results in genus two can be
found in [EMS], [HL], [Ca], [Mc1] and [Mc2].
Added in proof: Conjecture 1.2 is established in [Mc4].
2
Elementary moves
A form (X, ω) of genus two splits, in infinitely many ways, as a connected sum
of forms of genus one. In this section we define the intersection number of a pair
of splittings, and show they are related by a Dehn twist when their intersection
number is one.
Moduli spaces. We begin by recalling material from [Mc5]. Let ΩMg → Mg
denote the bundle of holomorphic 1-forms (X, ω), ω 6= 0, over the moduli space
of Riemann surfaces of genus g. Within the space ΩM2 of all forms of genus
two, we let
• ΩM2 (2) denote the closed stratum of forms with double zeros, and
• ΩM2 (1, 1), the open stratum of forms with simple zeros.
Connected sums. Let I = [0, v] = [0, 1] · v be the segment from 0 to v 6= 0 in
C, and let (Ei , ωi ) = (C/Λi , dz) ∈ ΩM1 be a pair of forms of genus one. When
I maps to an embedded arc under each projection C → Ei , one can slit along
these arcs and glue corresponding edges to obtains the connected sum
(X, ω) = (E1 , ω1 )#(E2 , ω2 ).
(2.1)
I
The connect sum is a form of genus two with a pair of simple zeros, coming from
the endpoints of the slits. To construct forms with double zeros, we also allow
the case where I projects to a loop in one torus Ei and remains embedded in
the other.
Splittings. Every form of genus two can be presented as a connected sum in
infinitely many ways [Mc5, Thm. 1.7], each of which we regard as a splitting of
(X, ω). The splitting (2.1) is uniquely determined by I, up to the ordering of
its summands.
6
Saddle connections. Let Z(ω) ⊂ X denote the zero set of ω; it is invariant
under the hyperelliptic involution η : X → X. A saddle connection is a geodesic
segment for the metric |ω|, with endpoints in Z(ω) but with no zeros in its
interior.
Given a splitting (2.1), the two sides of ∂(Ei − I) determine a pair of saddle
connections L, L′ on X such that η(L) = L′ . Conversely, by [Mc5, Thm. 7.3]
we have:
Theorem 2.1 Let L ⊃ Z(ω) be a saddle connection such that L 6= L′ = η(L).
Then (X, ω) splits along L ∪ L′ as a connected sum of tori.
Suitably oriented, the saddle connections L and L′ satisfy
Z
Z
Z
ω=
ω=
dz = v,
L
L′
I
and we let
[I] = [L] = [L′ ] ∈ H1 (X, Z(ω); Z)
denote the relative homology class they represent.
Intersection number. Given a pair of oriented saddle connections L, M on
(X, ω), the intersection number L · M is defined to be the algebraic number of
transverse crossings between L and M outside the zeros of ω. Note that all
crossings count with the same sign, and that L · M = −M · L.
We define the intersection number of a pair of distinct splittings
(X, ω) = (E1 , ω1 )#(E2 , ω2 ) = (F1 , η1 )#(F2 , η2 )
I
J
by
I · J = L · M + L′ · M = ((L + L′ ) · (M + M ′ ))/2,
where (L, L′ ) and (M, M ′ ) are the pairs of parallel saddle connections on (X, ω)
corresponding to I and J respectively.
J
I
Figure 2. Elementary move. The two dots mark the zeros of ω.
In the case I · J = 1, depicted in Figure 2, we can give a homological formula
relating the two splittings.
7
Theorem 2.2 Suppose (X, ω) ∈ ΩM2 (1, 1) admits a pair of splittings
(X, ω) = (E1 , ω1 )#(E2 , ω2 ) = (F1 , η1 )#(F2 , η2 )
I
J
satisfying ∂[I] = ∂[J] and I · J = 1. Choose symplectic bases (Ai , Bi ) for
H1 (Ei , Z) such that
[J] = [I] + B1 + B2
in H1 (X, Z(ω); Z). We then have
H1 (F1 , Z)
=
H1 (F2 , Z)
=
Z(A1 + B2 ) ⊕ ZB1 ,
Z(A2 + B1 ) ⊕ ZB2 .
(Here by a symplectic basis we mean Ai · Bi = 1.)
Proof. Let (L, M ) be a pair of oriented saddle connections on (X, ω) representing ([I], [J]) in relative homology and crossing positively at exactly one
point p 6∈ Z(ω). The pair (L, M ) is unique up to the action of the hyperelliptic
involution η.
We can regard Ei − I as a subsurface of X, whose closure Ti is a torus with
boundary L ∪ η(L). Let K ⊂ X − Z(ω) be a smooth, simple loop such that
η(K) = K, L and K cross transversally at p and nowhere else, and [K] = B1 +B2
in H1 (X, Z) when suitably oriented. Then K ∩ Ti and M ∩ Ti both represent
the class Bi ∈ H1 (Ti , ∂Ti ), i = 1, 2.
Let twK : (X, Z(ω)) → (X, Z(ω)) denote a left Dehn twist around K, fixing
Z(ω). It is then straightforward to verify that the arcs twK (L) and M are
isotopic, relative to their endpoints, using the fact that homologous loops on a
torus are isotopic. It follows that X splits along M ∪ η(M ) into a pair of tori
F1 , F2 satisfying
H1 (Fi ) = twK (H1 (Ei )).
Since the action of twK on H1 (X, Z) is given by
twK (x) = x + (x · K)K,
and Ai · (B1 + B2 ) = 1, a basis for H1 (Fi ) is as indicated above.
See [Mc5, §9] for another occurrence of this elementary move.
3
Prototypical splittings
In this section we briefly summarize the relationship between Teichmüller curves,
eigenforms and splittings in genus two. We then show every periodic splitting
of an eigenform is equivalent to a unique, concretely described model.
Teichmüller curves. Recall there is a natural action of GL+
2 (R) on the space
of holomorphic 1-forms ΩMg , and that the stabilizer SL(X, ω) of a given form
8
is a discrete subgroup of SL2 (R). The group SL(X, ω) is a lattice if and only if
(X, ω) generates a Teichmüller curve
f : V = H/ SL(X, ω) → Mg .
See e.g. [V1], [Mc5, Thm. 3.4].
Periodicity. Given (X, ω) ∈ ΩMg and s ∈ P1 (R), we have a foliation Fs (ω)
of (X, |ω|) by geodesics of slope s. The tangent space TFs (ω) ⊂ TX coincides
with the kernel of the harmonic 1-form ρ = Re(x + iy)ω, s = x/y. If all the
leaves of Fs (ω) are closed, we say Fs (ω) is periodic. By [V1, 2.4,2.11], we have:
Theorem 3.1 (Veech dichotomy) Suppose SL(X, ω) is a lattice. Then for
any slope s, the foliation Fs (ω) of X is either periodic or uniquely ergodic.
(In the uniquely ergodic case, one also knows there are no saddle connections
of slope s.)
Genus two. We say a splitting (X, ω) = (E1 , ω1 )#(E2 , ω2 ) is periodic if the
I
following equivalent conditions hold:
1. The foliation of (X, |ω|) by geodesics parallel to I is periodic;
2. I lies along a closed geodesic in each torus Ei ; and
3. ti v ∈ Λi for some ti > 0, i = 1, 2.
Theorem 3.2 Let (X, ω) be a form of genus two such that SL(X, ω) is a lattice.
Then every splitting of (X, ω) is periodic.
Proof. The foliation of (X, |ω|) by geodesics parallel to I has no leaves that
are dense in both E1 and E2 , so it cannot be uniquely ergodic; by the Veech
dichotomy, it must be periodic.
We also observe:
Theorem 3.3 If (X, ω) has two different periodic splittings, then the relative
and absolute periods of ω span the same rational subspace of C.
Eigenforms. Within the space ΩM2 of all forms of genus two, let
• ΩED denote the eigenforms for real multiplication by O D , and
• ΩED (2) and ΩED (1, 1), the eigenforms with one double and two simple
zeros respectively.
Each space above is invariant under the natural action of GL+
2 (R). The importance of eigenforms for the classification of Teichmüller curves comes from
[Mc5, Cor. 5.9]:
9
Theorem 3.4 If (X, ω) is a form of genus two and SL(X, ω) is a lattice, then
(X, ω) ∈ ΩED for a unique discriminant D.
Real multiplication and isogeny. We recall two further properties of eigenforms. First, if (X, ω) is an eigenform for real multiplication by OD , and D is
not a square, then
Per(ω) ⊗ Q ⊂ R2
√
is a 2-dimensional vector space over OD ⊗Q = Q( D). For the second property,
let us say that forms (E1 , ω1 ) and (E2 , ω2 ) of genus one are isogenous if there
is a surjective holomorphic map p : E1 → E2 such that p∗ (ω2 ) = tω1 for some
t ∈ R. Then by [Mc5, Thm. 1.8] we have:
Theorem 3.5 A form (X, ω) ∈ ΩM2 is an eigenform for real multiplication
iff every splitting of (X, ω) has isogenous summands.
Prototypes. We can now describe the periodic splittings of eigenforms (cf.
[Mc3, §3]).
Let us say a quadruple of integers (a, b, c, e) is a splitting prototype, of discriminant D, if it satisfies the conditions
D = e2 + 4bc,
0 < b,
0 ≤ a < gcd(b, c),
0 < c, and
c + e < b,
gcd(a, b, c, e) = 1.
The prototypical splitting of type (a, b, c, e) and width t ∈ (0, 1] is given by
(Xt , ωt ) = (E1 , ω1 )#(E2 , ω2 )
It
where It = [0, tλ], (Ei , ωi ) = (C/Λi , dz),
Λ1 = Z(λ, 0) ⊕ Z(0, λ), Λ2 = Z(b, 0) ⊕ Z(a, c),
√
and λ = (e + D)/2. The condition c + e < b in the definition of a prototype
is equivalent to λ < b, which insures that I projects an embedded arc in E2 .
(a,c)
(b,0)
tλ
λ
λ
Figure 3. Prototypical splitting.
10
The prototypical splitting can be expressed in geometric terms as (Xt , ωt ) =
(P, dz)/∼, where P ⊂ C is a polygon built from the period parallelograms for Λ1
and Λ2 as shown in Figure 3. The two parallelograms overlap along an interval
of length tλ corresponding to I. The equivalence relation identifies parallel edges
of P . As indicated in the figure, for 0 < t < 1 the vertices of P fall into two
classes, corresponding to the two zeros of ω, while for t = 1 all vertices of P are
equivalent, and ω has a double zero.
We refer to (Xt , ωt ) itself as the prototypical form of type (a, b, c, e) and
width t.
s
Orbits. Let ΩED
denote the splitting space, consisting of triples (X, ω, I) such
that (X, ω) ∈ ΩED splits as a connect sum
(X, ω) = (E1 , ω1 )#(E2 , ω2 ).
I
There is a natural action of
GL+
2 (R)
s
on ΩED
, and an equivariant projection
s
ΩED
→ ΩED ,
which is a local homeomorphism but not a covering map.
We can now state:
Theorem 3.6 Let D > 0 be a discriminant that is not a square. Then every
s
periodic splitting in ΩED
is equivalent, under the action of GL+
2 (R), to a unique
prototypical splitting.
s
Proof. Let (X, ω, I) ∈ ΩED
be a periodic splitting, with isogenous summands
(Ei , ωi ) = (C/Λi , dz), i = 1, 2, and with I = [0, v]. By periodicity, for i = 1, 2
there is a unique primitive vector ei ∈ Λi and ti ∈ (0, 1] such that v = ti ei .
Since the lattices Λ1 , Λ√2 rationally generate Per(ω) ⊗ Q ⊂ R2 , and the latter is
a vector space over Q( D), the ratio t1 /t2 must be irrational; in particular, we
can order the summands so that |e1 | < |e2 |.
Now let (Y, η) = (E1 , ω1 )#(E2 , ω2 ), where J = [0, e1 ]. Since the absolute
J
periods of η and ω agree, (Y, η) is also an eigenform [Mc5, Cor 5.6]. But now J
maps to a loop in E1 , so η has a double zero. By [Mc3, Thm. 3.3], there is a
unique prototypical splitting in the GL+
2 (R)-orbit of (Y, η, J), of type (a, b, c, e)
and width t = 1. Consequently the orbit of (X, ω, I) also contains a unique
prototypical splitting, namely that of type (a, b, c, e) and width t = t1 .
Corollary 3.7 Every Teichmüller curve generated by an Abelian differential of
genus two is also generated by a prototypical form.
Proof. Let f : V → M2 be a Teichmüller curve generated by (X, ω); then
(X, ω) is an eigenform and all its splittings are periodic, by Theorems 3.2 and
3.4. By the preceding result, the orbit of (X, ω) under GL+
2 (R) contains a
prototypical form.
11
4
Sifting
As we have just seen, the search for Teichmüller curves of genus two can be
reduced to the study of prototypical forms. But if SL(Xt , ωt ) is to be a lattice,
then all of its splittings must be periodic. In this section we will see such
periodicity imposes stringent conditions on the width t ∈ (0, 1).
Slopes. Fix a non-square discriminant D > 0, and let
√
K = Q( D) ⊂ R.
We emphasize that K is a real quadratic field with a fixed embedding into R.
Let (X, ω) ∈ ΩED be an eigenform for real multiplication by OD . Then
K∼
= OD ⊗Q acts on the rational homology of X, satisfying
Z
Z
ω
(4.1)
ω=k
C
k·C
for all C ∈ H1 (X, Q) and all k ∈ K. This action makes H1 (X, Q) into a vector
∼ 2
space over K, which
R we will denote simply by H1 (X) = K . The K-linear
function Iω (C) = C ω sends H1 (X) isomorphically to a dense subset of C.
Recall that the action of K on H1 (X) is self-adjoint with respect to the
intersection pairing; that is, it satisfies (kC) · D = C · kD. In particular, every
1-dimensional subspace K · C ⊂ H1 (X) is Lagrangian. The collection of all such
subspaces forms a projective line
PH1 (X) = (H 1 (X, Q) − {0})/K ∗ ∼
= P1 (K).
Given any set A ⊂ H1 (X), we define PA ⊂ PH1 (X) by
PA = {K ∗ · C : C ∈ A − {0}}.
Periods and periodicity. Any geometric splitting of an eigenform
(X, ω) = (E1 , ω1 )#(E2 , ω2 ),
I
I = [0, v], gives an algebraic splitting
H1 (X) = H1 (E1 ) ⊕ H1 (E2 ),
(4.2)
where H1 (Ei ) = H1 (Ei , Q). Since the summands above are symplectically orthogonal, there is a k ∈ K such that kH1 (E1 ) = H1 (E2 ) [Mc5, Lem. 6.3].
Taking the quotient of (4.2) by the action of K ∗ , we obtain
PH1 (E1 ) = PH1 (E2 ) ⊂ PH1 (X).
(4.3)
Note that PH1 (E1 ) is isomorphic to a copy of P1 (Q) inside PH1 (X) ∼
= P1 (K).
Now assume that the relative and absolute periods of ω span the same rational vector space in C. Then there is a unique class hIi ∈ H1 (X) such that
Z
Z
dz = v.
ω=
hIi
I
12
Theorem 4.1 A splitting (X, ω) = (E1 , ω1 )#(E2 , ω2 ) of an eigenform is periI
odic if and only if we have PhIi ∈ PH1 (E1 ).
Proof. As we saw in §3, periodicity is equivalent to the condition that v = ti ei
for some ti > 0 and ei ∈ Per(ωi ), i = 1, 2. This in turn is equivalent to
the condition hIi = ti ei in H1 (Ei ), since integration against ω maps H1 (X)
injectively to C. But the condition hIi = ti ei is equivalent to PhIi ∈ PH1 (Ei ),
and since PH1 (E1 ) = PH1 (E2 ), we need only check this condition for i = 1.
Prototypical setup. Now let (a, b, c, e) be
√ one of the finitely many splitting
prototypes of discriminant D, let λ = (e + D)/2, and let
(Xt , ωt ) = (E1 , ω1 )#(E2 , ω2 )
It
be the prototypical splitting of type (a, b, c, e) and width t ∈ (0, 1). Since the
summands in the algebraic splitting
H1 (Xt ) = H1 (E1 ) ⊕ H1 (E2 )
are independent of t, the vector spaces H1 (Xt ) are canonically identified as
t varies. Using this isomorphism, we obtain a symplectic basis (ai , bi )2i=1 for
H1 (Xt ) from the bases
Λ1
Λ2
= Z(λ, 0) ⊕ Z(0, λ)
= Z(b, 0) ⊕ Z(a, c)
Za1 ⊕ Zb1 ,
=
Za2 ⊕ Zb2
=
(4.4)
for the period lattices Λi = Per(ωi ) ∼
= H1 (Ei , Z).
Elementary moves. Consider a homology class C ∈ H1 (X, Z) that decomposes as C = B1 + B2 with Bi ∈ H1 (Ei , Z). We assume:
Bi is primitive and ai · Bi > 0 for i = 1, 2.
For each such class C = B1 + B2 , we will obtain a rationality condition that
must be satisfied by t if SL(Xt , ωt ) is to be a lattice.
To formulate this condition, choose classes Ai ∈ H1 (Ei , Z) satisfying Ai ·Bi =
1, so that the homology of X splits as a symplectic direct sum
H1 (X, Z)
=
L1
=
L2
=
L1 ⊕ L2 ,
with
Z(A1 + B2 ) ⊕ ZB1 ,
Z(A2 + B1 ) ⊕ ZB2 .
As in (4.3), we have PL1 = PL2 . Next, let
U (C) =
{t ∈ (0, 1) : (Xt , ωt ) splits along an interval J satisfying
∂[It ] = ∂[J], It · J = 1, and [J] = [It ] + C},
13
as in the statement of Theorem 2.2. It is easy to see that U (C) is a (possibly
empty) open interval in (0, 1). Finally, let
T (C) = {t ∈ P1 (K) : P(ta1 + C) ∈ PL1 }.
Note that T (C) = A(P1 (Q)) for some A ∈ PGL2 (K). (We include t = ∞ in
T (C) if a1 ∈ PL1 .) We then have:
Theorem 4.2 If t ∈ U (C), then the corresponding splitting of (Xt , ωt ) is periodic if and only if t ∈ T (C).
Proof. Let (Xt , ωt ) = (F1 , η1 )#(F2 , η2 ) denote the splitting corresponding to
J
t ∈ U (C). If this splitting is periodic, then the relative and absolute periods of
ωt span the same vector space over Q (as remarked in Theorem 3.3), and thus
t ∈ K. Moreover we have H1 (Fi ) = Li by Theorem 2.2, and
hJi = hIt i + C = ta1 + C
by the definition of U (C). By Theorem 4.1 we have PhJi ∈ PH1 (F1 ), which is
equivalent to P(ta1 + C) ∈ PL1 , and therefore t ∈ T (C). The converse is similar.
Triples of points. Since T (C) is a copy of P1 (Q), the result above sifts out
all but countably many values of t in the interval U (C).
Note that T (C) is determined by any three distinct points (t1 , t2 , t3 ) it contains; namely, it coincides with A(P1 (Q)) for the unique A ∈ PGL2 (K) sending
(0, 1, ∞) to (t1 , t2 , t3 ). In particular we have:
Theorem 4.3 If T (C) 6= T (D), then |T (C) ∩ T (D)| ≤ 2.
By varying the class C, the values of t for which SL(Xt , ωt ) can be a lattice can
often be reduced to a finite set.
We conclude with a concrete formula for T (B1 + B2 ).
Theorem 4.4 The set T (B1 + B2 ) contains the three points t1 , t2 , t3 ∈ K determined by the conditions
(t1 a1 − A1 ) ∧ B1 = (t2 a1 − A2 ) ∧ B2 = (t3 a1 + B1 ) ∧ B2 = 0,
where the wedge products are taken in
∧2K H1 (X) ∼
= K.
(Note that each solution ti is unique, because ai · Bi 6= 0.)
Proof. To show ti ∈ T (B1 + B2 ), it suffices to show that ti a1 + B1 + B2 lies in
K ∗ · L1 , or equivalently in K ∗ · L2 (since PL1 = PL2 ). For i = 3 the vanishing
of the wedge product above implies t3 a1 + B1 = k3 B2 for some k3 ∈ K, and
thus we have
t3 a1 + B1 + B2 = (1 + k3 )B2 ∈ K ∗ · B2 ⊂ K ∗ · L2
14
as claimed.
For i = 1 we note that a1 , A1 and B1 all lie in H1 (E1 , Q) ∼
= Q2 , and thus
the vanishing of the wedge product implies t1 a1 = k1 B1 + A1 with t1 , k1 ∈ Q.
We then have
t1 a1 + B1 + B2 = (1 + k1 )B1 + (A1 + B2 ) ∈ Q∗ · L1 ⊂ K ∗ · L1 .
Similarly, when the wedge product vanishes for i = 2, we can use the fact that
a1 = (λ/b)a2 to conclude that t2 a1 = t′2 a2 = A2 + k2 B2 , with t′2 , k2 ∈ Q. Then
t2 a1 + B1 + B2 = (1 + k2 )B2 + (A2 + B1 ) ∈ Q∗ · L2 ⊂ K ∗ · L2
as desired.
5
Aperiodicity
Let D > 0 be a non-square discriminant, let p = (a, b, c, e) be a splitting prototype of discriminant D, and let
(Xt , ωt ) = (E1 , ω1 )#(E2 , ω2 )
It
denote the prototypical splitting of type (a, b, c, e) and width t ∈ (0, 1). In this
section we will show:
Theorem 5.1 There is a t0 (p) > 0 such that (Xt , ωt ) admits an aperiodic
splitting for all t ∈ (0, t0 ).
Corollary 5.2 There is a dense, full-measure, open set U ⊂ ΩED such that
every form (X, ω) ∈ U admits an aperiodic splitting.
To prove Theorem 5.1, we first show explicitly that (Xt , ωt ) has many different splittings when t is small.
Theorem 5.3 Let Bi ∈ H1 (Ei , Z) be primitive vectors satisfying ai · Bi > 0
and K · B1 = K · B2 . Then we have 0 ∈ T (B1 + B2 ) and
(0, s) ⊂ U (B1 + B2 )
for some s > 0.
Proof. To show U (B1 + B2 ) contains (0, s), we will show (Xt , ωt ) splits along
a suitable interval J for all t sufficiently small.
To construct J, let αi be the unique geodesic path on Ei = C/Λi beginning
and ending at z = 0 and representing Bi ∈ H1 (Ei , Z). Since Bi is primitive, the
interior of αi is an embedded arc disjoint from the projection of I = [0, tλ] when
t is sufficiently small. The same is true for the nearby geodesic arc α′i beginning
15
at z = 0 and terminating at the other endpoint z = tλ of the projection of
I. Thus the four arcs α1 , α′1 , α2 , α′2 bound a quadrilateral Q on the connected
sum (Xt , |ωt |), whose short diagonal is one of the oriented saddle connections L
corresponding to I (Figure 4).
Since K · B1 = K · B2 , the arcs α1 and α2 are parallel, and therefore Q is
convex. Its long diagonal M then provides a second saddle connection, which
when suitably oriented satisfies
∂M
= ∂L,
[M ] = [L] + B1 + B2 , and
L·M
= 1
(since ai · Bi > 0). Moreover M is disjoint from η(L), and thus η(M ) 6= M .
By Theorem 2.1, the form (Xt , ωt ) admits a second splitting along an interval
J = [0, w] ⊂ C satisfying [J] = [M ]. The conditions above imply I · J = 1 and
[J] = [I] + B1 + B2 , and therefore t ∈ U (B1 + B2 ) for all t sufficiently small.
Finally, the condition K · B1 = K · B2 implies (ta1 + B1 ) ∧ B2 = 0 when
t = 0, and thus 0 ∈ T (B1 + B2 ) by Theorem 4.4.
E2
E1
Q
Figure 4. Construction of J when t is small.
Proof of Theorem 5.1. We will show there are classes Ci ∈ H1 (Xt , Z),
i = 1, 2, 3, such that U (Ci ) ⊃ (0, si ), si > 0, and such that
0, 1 ∈ T (C1 ),
0, 1/2 ∈ T (C2 ), and
0, b/λ ∈ T (C3 ).
For i = 1, let C1 = B1 + B2 where B1 = b1 and where B2 is the unique
primitive class in
(K · B1 ) ∩ H1 (E2 , Z) ∼
=Z
such that B2 · a2 > 0. Then U (C1 ) contains a neighborhood of zero by the
previous result, and 0 ∈ T (C1 ). Taking A1 = a1 , we find (a1 − A1 ) ∧ B1 = 0,
and thus 1 ∈ T (C1 ) by Theorem 4.4.
16
For i = 2, let C2 = B1 + B2 where B1 = a1 + 2b1 and where B2 is determined
by B1 just as before. Then we have U (C2 ) ⊃ (0, s2 ) and 0 ∈ T (C2 ). Moreover,
taking A1 = −b1 and t = 1/2, we have
(ta1 − A1 ) ∧ B1 = (a1 /2 + b1 ) ∧ (a1 + 2b1 ) = 0,
and therefore 1/2 ∈ T (C2 ).
Finally for i = 3, let C3 = B1 + B2 where B2 = b1 and where B1 is the
unique primitive vector in (K · B2 ) ∩ H1 (E1 , Z) satisfying a1 · B1 > 0. Then
we have U (C3 ) ⊃ (0, s3 ) and 0 ∈ T (C3 ). Moreover, taking A2 = a2 = ta1 with
t = b/λ, we have (ta1 − A2 ) ∧ B2 = 0 and therefore t ∈ T (C3 ).
Now let T0 = T (C1 )∩T (C2 )∩T (C3 ). Suppose |T0 | ≥ 3. Then T0 = T (Ci ) for
all i, and in fact T0 = P1 (Q), because it contains 0, 1 and 1/2. But b/λ ∈ T (C3 )
is irrational, so this is impossible.
Thus |T0 | ≤ 2. Therefore we can choose 0 < t0 < min(s1 , s2 , s3 ) such that
(0, t0 ) ∩ T0 = ∅. Then for any t ∈ (0, t0 ), there exists an i such that t 6∈ T (Ci ).
On the other hand we have t ∈ U (Ci ), so the corresponding splitting of (Xt , ωt )
is aperiodic by Theorem 4.2.
Width. To prove the Corollary, it is useful to introduce the function τ :
Ω EsD → R given by
(
R
|I|/| C ω| if the splitting along I is periodic, and
τ (X, ω, I) =
0
otherwise,
where C is the shortest closed geodesic on (X, |ω|) with the same slope as I.
The function τ is GL+
2 (R)-invariant, and satisfies τ (Xt , ωt , It ) = t for a
prototypical splitting; thus it extends the notion of width to general splittings.
It is straightforward to check that τ is upper semicontinuous; that is,
lim sup τ (Xn , ωn , In ) ≤ τ (X, ω, I)
if (Xn , ωn , In ) → (X, ω, I).
Proof of Corollary 5.2. Let PD be the finite set of splitting prototypes
p = (a, b, c, e) of discriminant D. By the preceding Theorem, for each p ∈ PD
there is a t0 (p) > 0 such that every prototypical form (Xt , ωt ) of type p with
t < t0 (p) has an aperiodic splitting. Let τ0 = min(t0 (p) : p ∈ PD ), and let
U = {(X, ω) ∈ Ω ED (1, 1) : τ (X, ω, I) < τ0 for some I.}
We claim that every (X, ω) in U has an aperiodic splitting. This is immediate
if (X, ω) splits along an interval I with τ (X, ω, I) = 0. Otherwise, X has a
periodic splitting with 0 < τ (X, ω, I) = t < τ0 . But then (X, ω) is GL+
2 (R)
equivalent to a prototypical splitting (Xt , ωt ) of type p and width t. Since
t < τ0 ≤ t0 (p), the form (Xt , ωt ) has an aperiodic splitting, so (X, ω) does as
well.
17
The set U is nonempty because it contains the prototypical forms of small
width, and it is open by semicontinuity of τ . Moreover U is GL+
2 (R)-invariant.
Since the action of GL+
(R)
is
ergodic
[Mc5,
Thm.
1.5],
U
is
a
dense open set
2
of full measure in ΩED .
6
Finiteness
In this section we establish:
Theorem 6.1 If D is not a square, then the set of (X, ω) ∈ ΩED such that
SL(X, ω) is a lattice falls into finitely many orbits under the action of GL+
2 (R).
Corollary 6.2 There are only finitely many Teichmüller curves in M2 with a
given non-square discriminant D.
For the proof we recall:
Lemma 6.3 Let Y be a single orbit in a closed, GL+
2 (R)-invariant set Z ⊂
ΩED . If Y is not open in Z, then int(Z) 6= ∅.
See [Mc5, Lem. 12.3].
Proof of Theorem 6.1. Suppose to the contrary that we can find an infinite
sequence of forms (Xi , ωi ) ∈ ΩED , i = 1, 2, 3, . . ., such that SL(Xi , ωi ) is a
lattice and such that
∞
[
Z=
GL+
2 (R) · (Xi , ωi )
1
is a countable union of disjoint orbits.
Let U ⊂ ΩED be the open, dense set of forms with aperiodic splittings
provided by Corollary 5.2. Since every splitting of every form (Xi , ωi ) is periodic,
we have Z ∩ U = ∅. Moreover we can normalize by the action of GL+
2 (R) so that
(Xi , ωi ) is a prototypical form of type pi = (ai , bi , ci , ei ) and width ti ∈ (0, 1].
There are only finitely many prototypes of discriminant D, so after passing to
a subsequence we can assume that pi = p is constant.
Next, note that inf ti > 0; otherwise one of the forms (Xi , ωi ) would have
an aperiodic splitting, by Theorem 5.1. Thus upon passing to a further subsequence, we can assume ti → t > 0, and ti 6= t for all i. Let Y ⊂ Z be the
GL+
2 (R) orbit of the prototypical form (Xt , ωt ) of type p and width t. Then Y
is not open in Z, and thus Z has nonempty interior by the Lemma above. But
this is impossible, because Z is disjoint from the open, dense set U .
18
7
Characterization of Teichmüller curves
In this section we establish:
Theorem 7.1 Let (X, ω) be a holomorphic 1-form of genus two. Then the
following are equivalent.
1. SL(X, ω) is a lattice.
2. For every slope s, the foliation Fs (ω) is either periodic or uniquely ergodic.
3. For every slope s, the foliation Fs (ω) is either minimal or periodic.
(Here Fs (ω) is minimal if every leaf disjoint from the zeros of ω is dense.) The
implication (2) =⇒ (1) is a converse to the Veech dichotomy in genus two,
while (3) =⇒ (2) shows:
Corollary 7.2 Let (X, ω) be a form of genus two. If every geodesic parallel to
a saddle connection is closed, then every other geodesic is uniformly distributed.
Cylinders and eigenforms. We begin with a dynamical characterization of
eigenforms. Recall that a cylinder for (X, |ω|) is a maximal open annulus C ⊂ X
foliated by closed geodesics of constant slope s.
Theorem 7.3 For any (X, ω) ∈ ΩM2 , the following conditions are equivalent:
1. (X, ω) is an eigenform for real multiplication.
2. Whenever (X, |ω|) has a cylinder of slope s, the foliation Fs (ω) is periodic.
A similar resulted was announced in [Ca]. For the proof, we will use:
Lemma 7.4 Let I = [0, v] ⊂ E = R2 /Z2 be an embedded segment on the square
torus. Then there are infinitely many slopes of simple closed geodesics L ⊂ E−I.
Proof. If I has rational slope, then it is contained in a simple closed geodesic
M ⊂ E, and for L we can take any simple geodesic passing through p ∈ M − I
with intersection number |L · M | ≤ 1. These geodesics represent infinitely many
elements of H1 (E, Z), and hence infinitely many slopes.
On the other hand, if I = [0, v] has irrational slope, then for any ǫ > 0 there
is an A ∈ SL2 (Z) ⊂ Diff(E) such that |A(I)| < ǫ. (This follows from the fact
that SL2 (Z) · v is dense in R2 .) Since A sends geodesics to geodesics, it follows
that the number of slopes of geodesics disjoint from I is arbitrarily large.
19
Proof of Theorem 7.3. We first show (1) implies (2). Suppose (X, ω) is an
eigenform, and C ⊂ X is a cylinder of slope s. By [Mc5, Thm. 7.4], there is
a splitting (X, ω) = (E1 , ω1 )#(E2 , ω2 ) such that a loop of ∂C is contained in
I
one of the two summands, say E1 − I. It follows that the geodesics of slope
s are closed on both (E1 , ω1 ) and (E2 , ω2 ), since the summands are isogenous.
Therefore those leaves of Fs (ω) that are contained entirely in one summand are
also closed. On the other hand, any closed geodesic of slope s on E1 that meets
I does so only once, since there is a parallel geodesic disjoint from I. Therefore
any leaf of Fs (ω) that crosses from E2 to E1 immediately returns to E2 in the
same position. Hence these leaves also close up on X, and therefore Fs (ω) is
periodic.
To show (2) implies (1), consider any splitting (X, ω) = (E1 , ω1 )#(E2 , ω2 ).
I
Order the summands so that I = [0, v] projects to an embedded arc in E1 .
Suppose a slope s ∈ P Per(ω1 ) is represented by a closed geodesic on (E1 , ω1 )
disjoint from I. Then (X, ω) has a cylinder C of slope s, carried by E1 . Assumption (2) implies all geodesics of slope s on (X, ω) are periodic, and thus
s ∈ P Per(ω2 ) as well.
Using the Lemma, we can obtain in this way infinitely many slopes shared
by P Per(ω1 ) and P Per(ω2 ). But P Per(ω1 ) and P Per(ω2 ) are simply copies
of P1 (Q) inside P1 (R), so once they share three points, they are equal. Their
equality implies the given splitting of (X, ω) has isogenous summands; and since
the splitting was arbitrary, Theorem 3.5 implies (X, ω) is an eigenform.
Lattices and periodic splittings.
generate Teichmüller curves.
Next we characterize eigenforms that
Theorem 7.5 Let (X, ω) be an eigenform. Then SL(X, ω) is a lattice if and
only if every splitting of (X, ω) is periodic.
Proof. If SL(X, ω) is a lattice, then every splitting of (X, ω) is periodic by the
Veech dichotomy (Theorem 3.1).
Now assume (X, ω) is an eigenform of discriminant D and every splitting of
(X, ω) is periodic.
Suppose D is not a square. Then there is an nonempty open set U ⊂ ΩED
consisting of forms with aperiodic splittings (Corollary 5.2). Therefore
Z = GL+
2 (R) · (X, ω) 6= ΩED ;
but this can only happen when SL(X, ω) is a lattice, by the classification of
orbit closures for the action of SL2 (R) on Ω1 M2 given in [Mc5, Thm. 1.2].
Finally, suppose D = d2 is a square. Then Λ = Per(ω) is a lattice in C, and
there is a degree d map p : X → E = C/Λ such that ω = p∗ (dz). Since every
splitting of (X, ω) is periodic, the relative and absolute periods of ω span the
same space over Q (Theorem 3.3), and therefore the difference c1 − c2 of the
critical values of p is a torsion point on E. Hence by [GJ], SL(X, ω) is a lattice
in this case as well, namely one commensurable to SL2 (Z).
20
Finally we characterize Teichmüller curves themselves.
Proof of Theorem 7.1. By the Veech dichotomy (Theorem 3.1 and the remark
that follows), (1) implies (2) and (3).
Now assume (2). Then every cylindrical direction is periodic, and hence
(X, ω) is an eigenform (Theorem 7.3). Moreover, every splitting of (X, ω) is periodic, since it cannot be uniquely ergodic (cf. Theorem 3.2). By the preceding
result, this periodicity implies SL(X, ω) is a lattice, and thus (2) implies (1).
The same reasoning applies with minimality replacing unique ergodicity, so
(3) also implies (1).
8
Curves with D = 5
The sifting technique introduced in §4 can be used to explicitly determine all the
Teichmüller curves with a given discriminant. In this section we demonstrate
the method by showing:
Theorem 8.1 There are exactly two Teichmüller curves with discriminant D =
5: one generated by the regular pentagon, and one generated by the regular
decagon.
ζ3
ζ4
M
L
N
ζ2
ζ
1
L’
Figure 5. The regular decagon.
The regular decagon. Let P ⊂ C be the regular decagon with vertices
{ζ i , 0 ≤ i ≤ 9}, where ζ = e2πi/10 is a primitive tenth root of unity (Figure
5). Identifying opposite sides of P , we obtain a holomorphic 1-form (X, ω) =
(P, dz)/∼ of genus two. Note that the vertices of P fall in to two equivalence
classes, and thus (X, ω) ∈ ΩM2 (1, 1). As shown by Veech, SL(X, ω) is a triangle
group of signature (5, ∞, ∞); in particular, (X, ω) generates a Teichmüller curve
in genus two [V1].
Since P has 10-fold symmetry, Jac(X) admits complex multiplication
√ by
Z[ζ] with ω as an eigenform. Noting that ζ + ζ −1 = γ, where γ = (1 + 5)/2
is the golden ratio, we see that Z[ζ] contains the real subring OD with D = 5.
Thus (X, ω) belongs to ΩE5 .
21
Lemma 8.2 The Teichmüller curve generated by the regular decagon is also
generated by the prototypical form of type (0, 1, 1, −1) and width t = (2 + γ)/5.
Proof. The foliation of P by horizontal lines descends to a periodic foliation
of (X, |ω|) by closed, horizontal geodesics, including four saddle connections labeled L, L′ , M and N in Figure 5. The hyperelliptic involution of X corresponds
to a 180◦ rotation of P , and thus η(L) = L′ . Therefore (X, ω) admits a periodic
splitting
(X, ω) = (E1 , ω1 )#(E2 , ω2 )
I
R
′
along L ∪ L , where I = [0, v] and v = L ω = ζ − ζ 4 .
Since p = (0, 1, 1, −1) is the only splitting prototype of discriminant D = 5,
(X, ω, I) is GL+
2 (R)-equivalent to a unique prototypical splitting (Xt , ωt , It ) of
type (0, 1, 1, −1) and width t. To compute the width, we note that t is simply
the ratio of |L| to the length of the shorter of the two closed horizontal geodesics
on E1 and E2 respectively. These geodesics are represented by L∪N and L∪M ;
since |M | < |N |, we have
t=
|L|
2+γ
ζ − ζ4
=
=
·
|L| + |M |
ζ − ζ4 + ζ2 − ζ3
5
C
Prototype (0, 1, 1, −1)
U (C)
T (C)
(0, 1, 0, 1)
(0, 1)
(0, 1, γ)
(1, 2, 1, 2)
(0, γ
−1
)
(0, 1/2, γ/2)
(0, 1, 1, 2)
(1/2, 1)
(1/2, 1, γ/2)
Table 6. Splitting information for D = 5.
Proof of Theorem 8.1. Let (Xt , ωt ) be the prototypical
form of type (a, b, c, e) =
√
(0, 1, 1, −1) and width t ∈ (0, 1]. Then λ = (−1 + 5)/2 = γ −1 , where γ is the
golden ratio.
By Corollary 3.7 and uniqueness of the splitting prototype, every Teichmüller
curve with discriminant D = 5 is generated by (Xt , ωt ) for some t. The case
t = 1 gives the regular pentagon (a form with a double zero), so we may assume
t < 1.
The homology basis (a1 , b1 , a2 , b2 ) given in (4.4) provides an isomorphism
H1 (Xt , Z) ∼
= Z4 . Let
C1 = (0, 1, 0, 1), C2 = (1, 2, 1, 2), C3 = (0, 1, 1, 2)
be homology classes with respect to these coordinates.
22
Figure 7. Pairs of splittings, near t = 0 and t = 1
By Theorem 5.3, U (C1 ) and U (C2 ) each contains a neighborhood of t = 0,
since they have the form Ci = (p, q, p, q). In fact, it is straightforward to check
that U (C1 ) = (0, 1), U (C2 ) = (0, γ −1 ) and U (C3 ) = (1/2, 1). Thus (Xt , ωt )
resplits along C1 and C2 when t is near 0, and along C1 and C3 when t is near
1 (Figure 7).
Theorem 4.4 allows one to explicitly compute three representative points in
T (Ci ) for i = 1, 2, 3. The results are summarized in Table 6. The remainder
of T (Ci ) ∼
= P1 (Q) consists of those x ∈ P1 (R) having rational cross-ratio with
respect to the three given points. Using this fact, we find:
T (C1 ) ∩ T (C2 )
T (C1 ) ∩ T (C3 )
= {0, ∞}, and
= {(2 + γ)/5, 1}.
Now suppose t ∈ (0, 1) and SL(Xt , ωt ) is a lattice. Then every splitting of
(Xt , ωt ) is periodic. By Theorem 4.2, if t lies in U (Ci ) then it must lie in T (Ci )
as well.
Since U (C1 ) ∩ U (C2 ) = (0, γ −1 ) and T (C1 ) ∩ T (C2 ) ∩ (0, 1) = ∅, we must
have t ≥ γ −1 . But then we have t ∈ U (C1 ) ∩ U (C3 ) = (1/2, 1). Since T (C1 ) ∩
T (C3 ) ∩ (0, 1) = {(2 + γ)/5}, only one value of t ∈ (0, 1) yields a lattice, namely
the value corresponding to the regular decagon.
9
Curves with D ≤ 400
As mentioned in the Introduction, the following conjecture implies that all Teichmüller curves of genus two are known.
Conjecture 9.1 Let D > 5 be a non-square discriminant, and let (X, ω) ∈
ΩED (1, 1) be an eigenform of discriminant D with simple zeros. Then SL(X, ω)
is not a lattice.
In this section we describe the proof of:
Theorem 9.2 The conjecture above holds for all D ≤ 400.
23
The proof is based on an algorithm that effectively determines all the Teichmüller curves with a given non-square discriminant D.
The algorithm. For ǫ > 0 we define (in the notation of §4)
T ǫ (C) = {t ∈ K : ta1 + C ∈ k(L1 ∪ L2 ) for some k ∈ K, |k| > ǫ}.
Since the lattices L1 and L2 are discrete, T ǫ (C) ∩ (0, 1) is finite. The definition
is arranged so that if t ∈ U (C) ∩ T (C), then t ∈ T ǫ (C) if and only if the
corresponding new splitting (Xt , ωt , J) is equivalent to a prototypical splitting
of width greater than ǫ.
To test the conjecture above for a given value of D, one may proceed as
follows.
1. Begin by enumerating the finitely many splitting prototypes PD of discriminant D.
2. For each p ∈ PD , find a set of open intervals Ii ⊂ (0, 1) and homology
classes Ci , Di , i = 1, . . . , np , such that
S
(a)
Ii contains a neighborhood of t = 0;
(b) Ii ⊂ U (Ci ) ∩ U (Di ) and
(c) T (Ci ) 6= T (Di ).
Here U (Ci ) and T (CS
i ) are taken with respect to the prototypical splitting
S
of type p. Let Tp = (Ii ∩ T (Ci ) ∩ T (Di )), and let Kp = (0, 1) − Ii .
3. If Tp and Kp are empty for all p ∈ PD , we are done — the conjecture is
verified for D.
S
4. Otherwise, let ǫ = inf( Tp ∪S
Kp ) > 0. For each p ∈ PD , find a finite set
of classes Ci such that Kp ⊂ U (Ci ). Then let Sp be the finite set
[
Sp = Tp ∪ (Kp ∩ U (Ci ) ∩ T ǫ (Ci )).
5. If Sp is empty for all p, again we are done — the conjecture is true for D.
S
6. Otherwise, let S = ({p} × Sp ) ⊂ PD × (0, 1). Consider S ∪ {∗} as the
vertex set of a finite graph G, initially with no edges.
For each (p, t) ∈ S, construct one or more new splittings (Xt , ωt , J) of the
prototypical form of type p and width t. If the new splitting (Xt , ωt , J) is
aperiodic, add an edge to G connecting (p, t) to {∗}. Otherwise, the new
splitting is GL+
2 (R) equivalent to a prototypical splitting of type and width
(q, s). If (q, s) ∈ S, connect it to (p, t) by an edge; otherwise, connect (p, t)
to {∗}.
7. If G is connected, the conjecture is verified.
24
8. Otherwise, choose one vertex from each component of G not containing
the vertex ∗, and let their union be the finite set S ′ . Then any Teichmüller
curve providing an exception to the conjecture is generated by a form of
type and width (p, t) ∈ S ′ .
Justification. To explain the algorithm, we first recall that every Teichmüller
curve is generated by a prototypical form (Corollary 3.7). Thus to verify the
conjecture, it suffices to determine for each p ∈ PD , the set of widths t ∈ (0, 1)
such that SL(Xt , ωt ) is a lattice. At the conclusion of step (2), we know (by
Theorem 4.2) that for SL(Xt , ωt ) to be a lattice, we must have t ∈ Tp ∪ Kp .
(The proof S
of Theorem 5.1 shows how to insure that a neighborhood of t = 0 is
covered byS Ii in step 2.)
Often Tp ∪ Kp is empty (see e.g. the case D = 8 below), so the conjecture
is already verified by step (3). If not, we can at least conclude in step (4)
that SL(Xt , ωt ) is never a lattice when t ≤ ǫ. Thus we can replace T (Ci ) with
T ǫ (Ci ), and thereby obtain a finite set S of pairs (p, t) accounting for every
possible Teichmüller curve of discriminant D.
To check that remaining steps, just observe that
(i) every splitting of a Teichmüller curve of discriminant D is equivalent to
one of type and width (p, t) ∈ S; therefore
(ii) the vertices (p, t) adjacent to ∗ in the graph G do not generate Teichmüller
curves; and
(iii) vertices in the same component of G label the same GL+
2 (R)-orbit, so if
one generates a Teichmüller curve, they all do.
As we saw in the previous section, when D = 5 the algorithm actually locates
the regular decagon and proves it generates the only exceptional Teichmüller
curve for this discriminant. Here are two more examples.
Prototype (0, 2, 1, 0)
C
U (C)
(0, 1, 0, 1)
(0, 1)
(1, 1, 1, 2)
(0, 1)
T (C)
√
(0, 1, 2)
√
(0, 1, 2/2)
C
(0, 1, 0, 1)
(1, 2, 1, 2)
(0, 1, 1, 2)
Prototype (0, 1, 1, −2)
U (C)
(0, 1)
√
(0, 2 − 2)
(1/2, 1)
T (C)
√
(0, 1, 1 + 2)
√
(0, 1/2, (1 + 2)/2)
√
(1/2, 1, (1 + 2)/2)
Table 8. Splitting information for D = 8.
D = 8. In this case there are only two splitting prototypes. With the classes
chosen in Table 8, we obtain Tp = Kp = ∅ for both types, establishing the conjecture without recourse to the graph G. (As in §8, we use the basis (a1 , b1 , a2 , b2 )
to describe classes C ∈ H1 (Xt , Z), and we give three representative points to
describe T (C).)
25
Prototype (0, 1, 3, −5)
U (C)
C
(0, 1, 0, 1)
(1, 2, 3, 2)
(0, 1, 1, 2)
(0, 1)
√
37)/2)
(0, (−5 +
(1/6, 1)
T (C)
√
(0, 1, (5 + 37)/6)
√
(0, 1/2, (5 + 37)/12)
√
(1/6, 1, (5 + 37)/12)
Prototype (0, 3, 3, −1)
C
U (C)
(0, 1, 0, 1)
(1, 2, 1, 2)
(0, 1)
√
(0, (3 + 37)/14)
T (C)
√
(0, 1, (1 + 37)/6)
√
(0, 1/2, (1 + 37)/12)
Table 9. Splitting information for D = 37.
D = 37. This case is more typical. There are nine splitting prototypes, of
which two are of particular note (see Table 9).
For the prototype p1 = (0, 1, 3, −5), we obtain a complete covering of (0, 1)
in step (2). However, if we use the classes C1 = (0, 1, 0, 1) and D1 = (0, 1, 1, 2)
to handle a neighborhood
of t = 1, we obtain Tp1 = {t1 } ⊂ T (C1 ) ∩ T (D1 ),
√
where t1 = (9 + 37)/22 ≈ 0.68558.
For the prototype
p2 = (0, 3, 3, −1), Tp2 is empty but we have Kp2 = [s, 1)
√
where s = (3 + 37)/14 ≈ 0.648769. This occurs because C1 = (0, 1, 0, 1) is
essentially the only class such that U (C1 ) contains a neighborhood of t = 1.
For every other p ∈ PD , we have Tp = Kp = ∅.
Since s < t1 , we obtain ǫ = s in step (4). This leads to
√
√
Sp2 = T ǫ (C1 ) = {t2 , u2 } = {(9 + 3 37)/14, (7 + 2 37)/11},
and finally to S = {(p1 , t1 ), (p2 , t2 ), (p2 , u2 )}. Luckily, it turns out that every
(p, t) ∈ S admits a splitting of type (q, s) with q 6∈ {p1 , p2 }. Thus all the vertices
of G are connected to ∗, and we obtain a proof of the conjecture for D = 37.
D ≤ 400. It is straightforward to automate the procedure just described, and
obtain a detailed, if lengthy, proof of Theorem 9.2. To conclude, we comment
on the general behavior of the algorithm for the 180 non-square discriminants
with 5 ≤ D ≤ 400.
First, the number of prototypes grows, albeit unevenly, as D does (Figure
10). (Note that |PD | is the same as the number of cusps of the Weierstrass
curve WD [Mc3, Cor 4.5].) The maximum number of prototypes, |PD | = 128, is
attained when D = 385. On the other hand, the coverings hIi , i = 1, . . . np i used
in step (2) of the algorithm remain rather small; the average value of np is 1.58,
and its maximum value is np = 6 (attained for p = (8, 9, 9, e) with e = −6, −7
and −8).
26
120
100
80
|P |
D 60
40
20
0
0
100
200
D
300
400
Figure 10. Number of splitting prototypes as a function of D.
Finally, we note that for 80 of the 180 values of D, the algorithm successfully
terminates at step (3). The value of |S| in step (4) is usually 1 and never more
than 5; the maximum is attained when D = 200.
It would be interesting to have a conceptual explanation for the algorithm’s
success.
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